Hausdorff dimension of Julia sets of unicritical correspondences

Abstract

We show that if β > 1 \beta >1 is a rational number and the Julia set J J of the holomorphic correspondence z β + c z^{\beta }+c is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of J J is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever q 2 > p q^2>p and β = p / q \beta =p/q in lowest terms.